B. R. Martin

  

301 + x pages.
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          Almost all physical scientists at some time come into contact with statistics. This is often initially during their undergraduate studies, but rarely is it via a full lecture course. Usually some statistics lectures are given as part of a general mathematical methods course, or as part of a laboratory course. Neither route is entirely satisfactory. The student learns a few techniques, typically unconstrained linear least-squares fitting and analysis of errors, but without necessarily the theoretical background that justifies the methods and allows one to appreciate their limitations. On the other hand, physical scientists, particularly undergraduates, rarely have the time, and possibly the inclination, to study mathematical statistics in detail. What I have tried to do in this book is therefore to steer a path between the extremes of a recipe of methods with a collection of useful formulas, and a detailed account of mathematical statistics, while at the same time developing the subject in a reasonably logical way. I have included proofs of some of the more important results stated in those cases where they are fairly short, but this book is written by a physicist for other physical scientists and there is no pretence to mathematical rigour. The proofs are useful for showing how the definitions of certain statistical quantities and their properties may be used. Nevertheless, a reader uninterested in the proofs can easily skip over these, hopefully to come back to them later. Above all, I have contained the size of the book so that it can be read in its entirety by anyone with a basic exposure to mathematics, principally calculus and matrices, at the level of a first-year undergraduate student of physical science.

Statistics in physical science is principally concerned with the analysis of numerical data, so in Chapter 1 there is a review of what is meant by an experiment, and how the data that it produces are displayed and characterized by a few simple numbers. This leads naturally to a discussion in Chapter 2 of the vexed question of probability – what do we mean by this term and how is it calculated. There then follow two chapters on probability distributions: the first reviews some basic concepts and in the second there is a discussion of the properties of a number of specific theoretical distributions commonly met in the physical sciences. In practice, scientists rarely have access to the whole population of events, but instead have to rely on a sample from which to draw inferences about the population, so in Chapter 5 the basic ideas involved in sampling are discussed. This is followed in Chapter 6 by a review of some sampling distributions associated with the important and ubiquitous normal distribution, the latter more familiar to physical scientists as the Gaussian function. The next two chapters explain how estimates are inferred for individual parameters of a population from sample statistics, using several practical techniques. This is called point estimation. It is generalized in Chapter 9 by considering how to obtain estimates for the interval within which an estimate may lie. In the final two chapters, methods for testing hypotheses about statistical data are discussed. In the first of these the emphasis is on hypotheses about individual parameters, and in the second we discuss a number of other hypotheses, such as whether a sample comes from a given population distribution and goodness-of-fit tests. This chapter also briefly describes tests that can be made in the absence of any information about the underlying population distribution.

All the chapters contain worked examples. Most numerical statistical analyses are of course carried out using computers, and several statistical packages exist to enable this. But the object of the present book is to provide a first introduction to the ideas of statistics, and so the examples are simple and illustrative only, and any numerical calculations that are needed can be carried out easily using a simple spreadsheet. In an introduction to the subject there is an educational value in doing this, rather than simply entering a few numbers into a computer program. The examples are an integral part of the text, and by working though them the reader’s understanding of the material will be reinforced. There is also a short set of problems at the end of each chapter and the answers to the odd-numbered ones are given in Appendix D. There are three other appendices: one on some basic mathematics, in case the reader needs to refresh their memory about details; another about the principles of function optimization; and a set of the more useful statistical tables, to complement the topics discussed in the chapters and to make the book reasonably self-contained.

In preparing a book, some errors and misprints are inevitable. Please notify me of any you may find. I would also be grateful for any other general comments. Any errors, misprints and comments will be listed here.